A Conjecture of Warnaar-Zudilin from Deformations of Lie Superalgebras

TL;DR

This paper proves a set of $q$-series identities related to superconformal field theory by deforming affine vertex operator superalgebras, connecting characters of principal subspaces with those of simple affine superalgebras.

Contribution

It introduces a novel deformation approach of affine vertex operator superalgebras that confirms conjectured $q$-series identities and links principal subspace characters to superalgebra characters.

Findings

Proved Warnaar-Zudilin conjectured $q$-series identities.

Established a deformation method connecting superalgebra characters.

Filled a gap in the understanding of principal subspace characters.

Abstract

We prove a collection of $q$-series identities conjectured by Warnaar and Zudilin and appearing in recent work with H. Kim in the context of superconformal field theory. Our proof utilizes a deformation of the simple affine vertex operator superalgebra $L_k(\mathfrak{osp}_{1|2n})$ into the principal subsuperspace of $L_k(\mathfrak{sl}_{1|2n+1})$ in a manner analogous to earlier work of Feigin-Stoyanovsky. This result fills a gap left by Stoyanovsky, showing that for all positive integers $N$, $k$ the character of the principal subspace of type $A_N$ at level $k$ can be identified with the (super)character of a simple affine vertex operator (super)algebra at the same level.